The Monty Hall Problem
Named for the long-time host of the television game show Let’s Make A Deal, the Monty Hall problem has been a popular example in probability since the 1970’s because it illustrates in a compelling way how the laws of probability often confound our intuition. It is sometimes called a paradox because the answer can be difficult to accept, but it is not actually a paradox.
The problem itself was a common feature on the show: a contestant is shown three curtains on the stage, and told that behind exactly one of them is a fabulous prize, such as a luxury car, and the contestant has only to pick the right curtain to get it.
Once the contestant picks a curtain (Curtain 2, say), the host then opens one of the other curtains (that the contestant hasn't picked), and invariably this reveals a joke prize, such as a farm animal.
At this point the host asks the contestant whether he or she wishes to stay with the first choice, or change to the other unopened curtain. If you were the contestant, would you stick with Curtain 2, or switch now to Curtain 1? Take a moment to consider your answer.
* * * * *
Most people—including those on the show—stick with their original choice, because their intuition tells them there is an even chance between the remaining curtains, and most of us are not prone to change our decisions without a good reason. But intuition is wrong in this case: the odds of the prize being behind each of the remaining curtains are not even, not at all. In fact, by switching after the goat is revealed, the contestant doubles his or her probability of winning.
The simple explanation is that the contestant had only a 1/3 chance at guessing the correct curtain to begin with, and so there is a 2/3 chance that the prize is behind one of the curtains not picked. That the host later reveals what is behind one of these doesn't alter this initial condition; there is still a 2/3 chance the prize is behind the unpicked curtain, and only a 1/3 chance it is behind the curtain already picked.
Many people do not find themselves completely convinced by this explanation, so let us examine a picture that analyzes the sample space into the simple events of winning and losing, and consider exactly where the probabilities lie.
This is actually a very simple tree if you only pay attention to the branches that have any probabilities attached, namely the first left and right branch. The player has a random chance of selecting one of the two wrong curtains with probability 2/3, and of selecting the correct curtain with probability 1/3.
Nothing later changes these probabilities: nothing that occurs subsequent to this first choice is in any sense random, because the host is guaranteed to not reveal the location of the prize. Therefore, the player who always elects to switch after the reveal will win the prize two games out of three, on average, while the player who never switches will win only once out of three.
The inability of our intuition to correctly assess probabilities can affect us in adverse ways—and not just in game shows. To properly understand comparative risk, particularly in our behaviors and lifestyle choices, we should always welcome an opportunity to hone our understanding of the mathematics of probability.